The sum of the first n terms of an arithmetic sequence is n/2 (4n+20).
Find the first term and the common difference of the above sequence.
To find the first term and the common difference of the arithmetic sequence, we can use the formula for the sum of an arithmetic sequence and the given information.
The formula for the sum of the first n terms of an arithmetic sequence is:
Sn = (n/2)(a + l)
where:
Sn is the sum of the first n terms,
n is the number of terms,
a is the first term, and
l is the last term.
In this case, we are given that:
Sn = n/2 (4n + 20)
To find the first term, we need to find the value of a.
Let's start by simplifying the given sum formula:
n/2 (4n + 20) = (2n/2)(2n + 10) = n(2n + 10) = 2n^2 + 10n
Now, since this sum formula represents an arithmetic sequence, we know that the sum of the first n terms can be expressed as the product of the average of the first and last terms and the number of terms:
Sn = (n/2)(a + l)
Comparing this to the simplified formula we obtained earlier, we can equate the expressions for Sn:
2n^2 + 10n = (n/2)(a + l)
Simplifying further, we have:
2n^2 + 10n = (an + al) / 2
Multiply both sides of the equation by 2:
4n^2 + 20n = an + al
Since the given sequence is arithmetic, we know that the difference between consecutive terms is constant. Let's represent the common difference as d.
So, we have:
an = a + (n - 1)d
al = a + (n)d
Substituting these values into the equation, we get:
4n^2 + 20n = a + (n - 1)d + a + (n)d
Now, we can simplify the equation further:
4n^2 + 20n = 2a + 2nd
Comparing the coefficients of n^2 and n, we get:
4 = 0
20 = 2d
From the second equation, we can solve for d:
2d = 20
d = 10
Therefore, the common difference (d) of the sequence is 10.
To find the first term (a), we can substitute the value of d into one of the previous equations. Let's use the equation al = a + (n)d:
a + (n)d = a + (n - 1)d
a + n(10) = a + (n - 1)(10)
10n = 10(n - 1)
10n = 10n - 10
0 = -10
Since the equation leads to an inconsistency (0 = -10), there is no unique solution resulting from the given equation for the first term. It is not possible to determine the first term of the arithmetic sequence based on the given information.
a = 2n
and (n-1)d = 20
so (2n-2)d = 40
(a-2)d = 40
So we could pick
a=3, d=40
check:
3,43,83,...3+40(n-1) = n/2 (6+(n-1)40)
= 3n + 20n^2 - 20n = n(20n-17)
But there are many other possibilities.